An modified relax-and-fix algorithm for the multi-level lot-sizing problem with replaceablility

This paper considers the multi-level capacitated lot-sizing problems, especially pays attention paid to the replace-ability among items. We establish two nonlinear programming models for the problem: the inventory and lot-sizing model (ILSR) and the simplified facility location model (SFLR) based on different variable definitions with the objective to minimize the system-wide costs, including inventory-holding cost, backlog cost and machine setup cost. For small-scale problems, the commercial solver can solve the problems to optimality. For large-scale problems, we propose a modified relax-and-fix algorithm for exact algorithms encounters computational difficulty due to the NP-hardness of the problem. Numerical experiments are conducted to test the models and the designed algorithm, and the results show that the SFLR model performs better than the ILSR model and the designed algorithm works well.